Post on 17-Apr-2018
REVIEW (SUMMARY)
¢ Oppenheimer Volkhoff limit: upper limit to mass of neutron star remnant more than 1.4 𝑀↓⊙ à neutron degeneracy
¢ Supernova à extremely dense neutron star ¢ If mass above 2 – 3 𝑀↓⊙ , the star collapses into black
hole ¢ Chandrasakhar limit – upper mass limit of star
remnant less than 1.4 𝑀↓⊙ à electron degeneracy ¢ Outer layers blow off – planetary nebulae ¢ Core collapses further – white dwarf
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NEWTON’S UNIVERSE
Newton thought the Universe to be ¢ Infinite ¢ Uniform ¢ static
¢ If this were true, the night sky should be bright – where ever you look should be a star
CGR\Why is it Dark at Night_ - YouTube [720p].mp4 ©
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OLBER’S PARADOX
¢ In a given volume of space imagine an average number of stars N with density 𝜌
¢ Imagine stars in a thin shell of thickness d at a distance R
¢ Volume of shell: 4𝜋𝑅↑2 𝑑 𝑎𝑛𝑑 # of stars N in shell = n 4𝜋𝑅↑2 𝑑
¢ But brightness 𝑏= 𝐿/4𝜋𝑅↑2 ¢ Volume of shell increases
with 𝑅↑2 , but luminosity decreases with 1/𝑅↑2
¢ Total brightness we receive from any shell should be
¢ 𝐵=𝑁𝑏 ¢ 𝐵=𝑛4𝜋𝑅↑2 𝑑× 𝐿/4𝜋𝑅↑2 =𝑛𝑑𝐿
R
Thickness 𝑑 #Stars ∞𝑅↑2
∞1/𝑅↑2
¢ Overall brightness: add up all shells
¢ Since assumed Universe is infinite, there are an infinite # of shells
¢ There should be an infinite bright night sky
Amount of light received
from shell does not
depend upon how far away
shell is
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POSSIBLE SOLUTIONS
1. Universe not infinite or it is not uniform à but there is no edge à it must be infinite
2. Light is absorbed by something à s.th. Should warm up and radiate energy à not observed
3. Not static à we know it is expanding from red and blue shift à it is expanding now, but what does it imply about the past?
Is the Universe
Infinite and Uniform?
Has all light reached us?
Is the Universe
static?
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¢ Universe was created ~ 13 – 15 billion years ago
¢ We only could see light that was less than 15 billion ly away
¢ If we receive light from a finite # of stars, the night sky would be dark
Has the known
Universe changed?
Big Bang
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BIG BANG MODEL
¢ Doppler effect causes redshift for sources moving away àlonger λ à lower f
¢ Light sources are compared in their spectrum against the spectrum of a stationary hydrogen source
¢ Galaxies show redshift à they move away from each other à they must have been much closer together in the past
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AT THE BEGINNING…
¢ …there was a point of infinite density called singularity
Big Bang Explosion
• Time and space was created some 10 – 20 billion years ago
• It is the space in which galaxies are situated that expands
• Balloon analogy, but à no outside à no center
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QUESTIONS
¢ What is outside the Universe? ¢ What was there before the Big Bang?
Questions are meaningless – they can’t be asked
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REDSHIFT
¢ Due to photons traveling through space
¢ Space expands causing the λ of photons to expand
¢ The longer a photon travels, the more its λ will increase
¢ The photons originating from the furthest distance will be redshifted most
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DICKE AND PEEBLES IN 1960 Proposal: CGR\Cosmic Origins clip_ Arno Penzias and the Expanding Universe - YouTube [720p].mp4 ¢ During early Universe sufficient
high temperature to produce helium by fusion (24% of the Universe is helium)
¢ High energy photons were produced ¢ They have a BB spectrum
corresponding to the temperature at that time
¢ As Universe expanded and cooled, photon spectrum would also cool and max λ would change in accordance with Wien’s Law
¢ Today they should have a max λ corresponding to 3K
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PENZIAS AND WILSON - 1965
• No matter where they pointed their aerial, it picked up a steady continuous background radiation
• All space is filled with radiation corresponding to a BB spectrum of 2.76K
• Evidence for the Bing Bang Model
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BIG BANG MODEL
¢ Space is expanding and within space distribution of galaxies is not uniform
¢ Universe is not uniform or static ¢ Resolution to Olber’s paradox
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EXAMPLE ¢ This question is about the Big Bang model
and red-shift. ¢ (a) Describe what is meant by the Big Bang
model. Solution ¢ space and time originated from a single point in a large explosion
¢ (b) In the 1960s, Penzias and Wilson discovered a uniform cosmic background radiation (CMB) in the microwave region of the electromagnetic spectrum.
(i) Explain how the CMB is consistent with the Big Bang model. Solution ¢ temperature of the universe immediately after the Big Bang was very high;
as it expanded it cooled down; the wavelength of the CMB corresponds to a temperature consistent with this cooling down
¢ (ii) State why the red-shift of light from galaxies supports the Big Bang model.
¢ indicates that the universe is expanding;
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EXAMPLE
This question is about Olbers’ paradox. ¢ (a) Newton assumed that the universe is static and
that the stars are uniformly distributed. State one further assumption of the Newtonian universe.
Solution ¢ universe is infinite
¢ (b) Explain how Newton’s assumptions led to Olbers’ paradox.
Solution ¢ number of stars in shell increases as 𝑅↑2
intensity decreases with 1/𝑅↑2 ; hence brightness of shell is constant; adding all shells to infinity; sky would be uniformly bright;
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HUBBLE’S LAW (1929)
¢ Estimating the distance to galaxies, Hubble showed that the recession speed is proportional to the distance of galaxies to Earth
¢ Hubble’s Law 𝑣= 𝐻↓0 ×𝑑
¢ v = recession speed ¢ d = distance to galaxy ¢ 𝐻↓0 = Hubble’s constant
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VALUE FOR 𝐻↓0 ¢ An accurate value for Hubble’s constant is
difficult to measure ¢ Units are important:
v = km 𝑠↑−1 d = Mpc 𝐻↓0 =𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1
¢ From the gradient of the graph of Hubble’s data the original value for 𝐻↓0 = 500 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1
¢ Modern values state 𝐻↓0 = 70 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 Still contains uncertainty
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NEWEST VALUES FOR 𝐻↓0
December 2012 ¢ NASA’s Wilkinson microwave
Anisotropy Probe (WMAP) gave a value of
𝐻↓0 = (69.32±0.80) 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1
March 2013 ¢ Planck mission provided a
value of 𝐻↓0 = (67.80±0.77) 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1
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EXAMPLE
¢ The value of the Hubble constant 𝐻↓0 is accepted to be in the range of 60 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 to 90 𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 . A) state and explain why it is difficult to determine a precise value for 𝐻↓0
Solution ¢ The further galaxies away, the
more difficult it is to determine the distance accurately
¢ You need a standard candle to accurately measure its luminosity
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EXAMPLE
¢ B) State one reason why it would be desirable to have a precise value of 𝐻↓0
Solution ¢ We could get an accurate value of the rate of
expansion of the Universe ¢ We could get an accurate value of the distance to
the galaxy We could find an accurate value for
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LIMITATIONS OF HUBBLE’S LAW
¢ Standard candle ¢ Up to 60Mpc
¢ 250Mpc < d < 900Mpc
¢ 60Mpc < d < 250Mpc
¢ 900Mpc < d < 1000Mpc
Cepheid Variable Red and blue super giants
Brightest globular clusters Supernova
Remember: as d ↑ error in measurement ↑
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AGE OF THE UNIVERSE AND 𝐻↓0 ¢ Assume the Universe is expanding at constant
rate. Two galaxies are separated by a distance d. How long did it take to get this distance apart?
Solution ¢ Let the age or expansion time of the
Universe be T, the Hubble time will be 𝑇= 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒/𝑠𝑝𝑒𝑒𝑑
¢ T = 𝑑/𝐻↓0 𝑑 ¢ 𝑇= 1/𝐻↓0 ¢ This means the Universe has an age changing
depending on 𝐻↓0
d
But v = 𝐻↓0 𝑑
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CALCULATING THE AGE OF THE UNIVERSE
¢ Assume 𝐻↓0 =70𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1 . ¢ 𝑇= 1/70𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1
¢ 𝑇= 10↑6 𝑥9.46𝑥10↑15 𝑥3.26/70𝑥10↑3
¢ 𝑇= 3.1𝑥10↑22 /70𝑥10↑3 = 4.4 𝑥10↑17 𝑠 ~1.4𝑥10↑10 𝑦𝑒𝑎𝑟𝑠
1ly=9.46x10↑15 𝑚 1pc = 3.26ly
14 billion years
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ASSUMPTION
¢ Galaxies and Earth are moving at constant speed of light c WRT each other
¢ There is nothing in their way to slow them down
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HUBBLE TIME
¢ Assume à galaxies have traveled at constant speed
¢ If rate of expansion is slowing down, actual age of the Universe would be less than the Hubble time
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THE ACCELERATING UNIVERSE ¢ To trace expansion history of the Universe, a
supernova needs to be used as standard candle ¢ A 1990 supernova was used to find distance for
about 1000Mpc ¢ It was found that at redshifts of 0.5 the
supernova was fainter than expected for a constant expansion
¢ Conclusion: it must accelerate as it moves further away
¢ Energy density that acts in opposition to gravity causes expansion to accelerate
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BUT…..
¢ ….Supernovae at redshifts between 0.5 and 1 were decelerating
¢ Whether Universe will expand forever or will collapse is something that cosmologists are trying to answer in terms of dark energy
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REDSHIFT: LENGTHENING OF WAVELENGTH
¢ Relative motion of source and observer – Doppler effect
¢ Relativistic time dilation
¢ A photon losing energy as it climbs out a gravitational field
What can cause redshift?
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COSMOLOGICAL REDSHIFT Z
Only used for redshift caused by overall expansion of space ¢ 𝑧= λ↓𝑜𝑏 − λ↓𝑟𝑒𝑠𝑡 /λ↓𝑟𝑒𝑠𝑡 ¢ 𝑧= ∆λ/λ↓0 ~ 𝑣/𝑐 v= velocity of source c = speed of light
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EXAMPLE
¢ The wavelength of the blue line in the spectrum of atomic hydrogen as measured in the lab on Earth is 486nm. In the spectrum from a distant galaxy the same blue line measures 498nm. Estimate the recession speed of the galaxy from Earth
Solution ¢ ∆λ=498 −486=12 𝑛𝑚 ¢ ∆λ/λ↓0 = 𝑣/𝑐 ¢ 𝑣= ∆λ/λ↓0 ×𝑐= 12/486 ×3.0× 10↑8 =7407407𝑚𝑠↑−1 ¢ 𝑑= 𝑣/𝐻↓0 ¢ 𝑑= 7407𝑘𝑚𝑠↑−1 /65 =113.95~114𝑀𝑝𝑐
Assume 𝐻↓0 =65𝑘𝑚𝑠↑−1 𝑀𝑝𝑐↑−1
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THE COSMIC SCALE FACTOR R
¢ As the Universe expands, all distances are stretched by the cosmic scale factor R
¢ If radiation was λ↓0 when emitted, but λ when it was detected, the cosmic scale factor would have changed from 𝑅↓0 to R
¢ Space has stretched by an amount ∆𝑅 in the same time λ has stretched by ∆λ
¢ Hubble’s law holds because space is expanding, not galaxies receding
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SCALE FACTOR R
¢ 𝑧= ∆λ/λ = ∆𝑅/𝑅 = 𝑅− 𝑅↓0 /𝑅↓0 = 𝑅/𝑅↓0 −1 ¢ 𝑧= 𝑅/𝑅↓0 −1 ¢ R = present scale factor ¢ 𝑅↓0 = scale factor when radiation was emitted ¢ 𝑅/𝑅↓0 = factor by which Universe has expanded
since time presently observable radiation was emitted
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EXAMPLE
¢ For a Quasar of redshift 4, we are seeing it as it was when the Universe was how big?
¢ Solution ¢ 𝑧+1=𝑅/𝑅↓0 ¢ 1/𝑧+1 = 𝑅↓0 /𝑅 ¢ 𝑅↓0 = 1/4+1 𝑅= 1/5 𝑅
The Universe was 1/5 of its present value
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EXAMPLE ¢ A distant Quasar is detected to have a redshift value z of 5.6
a) find the speed of the Quasar WRT Earth b) estimate the ratio of its current size to the Universe’s size when the Quasar emitted the photons that were detected
Solution ¢ a) 𝑧= ∆λ/λ↓0 = 𝑣/𝑐 =5.6
v = 5.6c ¢ b) 𝑅/𝑅↓0 −1=5.6 ¢ 𝑅/𝑅↓0 =5.6+1=6.6 ¢ 𝑅↓0 /𝑅 = 1/6.6 ¢ 𝑅↓0 =0.15𝑅
The Universe
was 15% of its current
size
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